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"\n", "Task 1: \n", " \n", "1. Identify the main physical processes that are at play in this problem, and get familiar with the equations available to describe them.\n", "2. Write an equation (on paper) to define failure of the dike. You can use the code provided to check your answer.\n", "3. Now that you have an equation, identify which random variables have a positive or negative effect on the performance of the dike.\n", "4. Which environmental variable governs the dike design in this situation? Wave height or wind setup?\n", "
\n", "\n", "Task 2: \n", " \n", "1. Define the distributions of each random variable using `scipy.stats` and get familiar with their characteristics\n", "2. Which random variables have the most significant variability?\n", "3. Identify how each random variable influences the equations defined in the deterministic part above.\n", "4. Which random variable do you think will govern the dike design in this situation?\n", "
\n", "\n", " Solution:\n", " \n", "1. See code below.\n", "2. See computations and summary table below. Note that for the non-Gaussian distributions the location and scale parameters are **not** the same thing as the mean and standard deviation! However, in this case, the numerical values are not that different. In the summary table, you have the coefficient of variation (c.o.v.) which is the quotient between the standard deviation and the mean of a random variable, giving a dimensionless metric of its variability. If we compare the c.o.v. of the different variables, we can see that the highest variability is that of the significant wave height. But does that mean it will have the biggest impact on the dike design? Not necessarily!\n", "3. Recall from Part 1 that the random variables all combine to produce an effective water level at the dike. We can't tell from the distributions of the random variables themselves if the wind speed or the wave height would increase the effective water level observed at the dike more _given the probability distributions associated with them._ To understand that we need to somehow combine the likelihood of high values of these \"load\" variables along with the mechanical response of the dike (i.e., plug the values into the function from Part 1!). We will do this in Part 3.\n", "4. Same as above.\n", "
\n", "\n", "Task 3: \n", " \n", "1. Using the Python functions provided, identify conditions where the dike (in its \"current\" form) is unsafe (using deterministic values only). Describe these critical conditions using a few values and figures.\n", "2. Compute the probability that dike is unsafe. Does it meet the safety criteria (typically, a 1% probability of failure)?\n", " \n", "
\n", "\n", " Solution (part 1):\n", " \n", "We already have a function to quantify the demand, or \"load,\" on the system: $Z_d$. The survival or \"resistance\" of the dike is the crest height, $Z_{crest}$, so we simply need to check whether the calculated load exceeds the capacity (the crest height).\n", "\n", "The \"Monte Carlo Simulation\" part comes in because we would like to do this check for many realizations of the random variables (all the possible values and combinations), to estimate the failure probaiblity as the fraction of realizations where the dike has \"failed.\" Since the random variables are assumed to be statistically independent (note that in reality $H_s$ and $u$ are usually not independent!), we can sample from each distribution separately, then plug the values into our function to obtain an empirical distribution of $Z_d$. Then the failure probability is simply $P[Z_d>Z_{\\mathrm{crest}}]$.\n", "\n", "This is illustrated in the code below.\n", "
\n", "\n", " Solution (part 2):\n", " \n", "As we can see, the calculated probability far exceeds the allowable failure probability of 1%, so we must do something to improve this dike!\n", "\n", "
\n", "\n", "Task 4: \n", " \n", "1. Reconsider the deterministic and random variables and identify if and how each of them could be changed to improve the system. You should be able to think of explicit examples for what you can do in \"real life\" and explain the quantitative effect on each variable. For the random variables, you should explicitly consider whether you can influence one of three things: the mean, standard deviation or type of distribution. You should also explain whether a measure is practical. For example: _making the inlet more shallow by placing dredge spoils in it would reduce the mean value of water depth and would impact the runup height by ... ... ...; however, this is probably very expensive and not a very practical idea!_\n", "2. Make a recommendation for how you can improve the system to meet the safety criteria. Explain the quantitative effect on the system (as described above) and provide quantitative confirmation that in these conditions the dike is safe.\n", " \n", "
\n", "\n", " Solution:\n", "\n", "There are many approaches to this, so you are encouraged to use the code yourself to explore various options. The main approach is to change the deterministic parameters and the location and scale of the random variables to see the quantitative effect on probability. Observations should include:\n", "- you can have achieve big improvements by changing $u$ and $H_s$, however, since these are natural phenomenon, it is difficult to influence them. One practical measure could be installing something in the inlet to reduce wave heights.\n", "- a similar observation can be made about the water and bottom elevations: these variables have an influence on the calculated probability, but are difficult to make changes in practice\n", "- the most \"obvious\" or \"practical\" solutions would be to raise the dike or decrease the slope (this only impacts the wave runup). The choice for which one is selected would probably come down to the cost of materials (i.e., how does the cross-sectional area change?). To determine the needed crest height, we can plot the calculated distribution of the loading $Z_d$ and see what is the value associated with an exceedance probability of 1%. In the plot below, it can be seen that a crest level of approximately 10.2m would provide the required safety level. \n", " \n", "See code below as example of the computed options.\n", "\n", "
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